= First homotopy group above the dimension of a sphere
{title2=$\pi_{n+1}(S^n)=0\ (n=1),\ \mathbb Z\ (n=2),\ \mathbb Z/2\ (n\geq3)$}
The <universal cover> of $S^1$ gives the first case. The <Hopf fibration> $S^1\to S^3\to S^2$ identifies $\pi_3(S^2)$ with $\pi_3(S^3)=\mathbb Z$. Starting from $\pi_4(S^3)=\mathbb Z/2$, the <Freudenthal suspension theorem> makes every subsequent <suspension> in these degrees an isomorphism. The stable range does not include the first step from $\pi_3(S^2)$ to $\pi_4(S^3)$, so the integer group at $n=2$ is not a contradiction.
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